He+ (2015). |
There are 3 generations of fermion (\(e\), \(\mu\), \(\tau\)), but only 1 generation of bosons.
In string theories with fermions living on a compact internal manifold whose tangent bundle is, as here, the gauge bundle, the number of generations (number of massless gauge-multiplet solutions of the Dirac equation, also known as the index of the Dirac operator \(\boldsymbol{\gamma}^m D_m\)) equals \(\tfrac{1}{2} |\chi|\), half the absolute value of the Euler characteristic \(\chi\) of the compact manifold (Atiyah-Singer index theorem).
The first Calabi-Yau three-fold discovered with \(\tfrac{1}{2} |\chi| = 3\) was a core ingredient of the influential proposal by Candelas+ (1985) to compactify 10d \(\textrm{E}_8 \times \textrm{E}_8\) heterotic string theory to 4d on a 6d Calabi-Yau three-fold.
There have been extensive investigations of Calabi-Yau three-folds, and some investigation of Calabi-Yau four-folds, but relatively little on Calabi-Yau five-folds. There is a plethora of Calabi-Yau manifolds, although only 9 known three-folds with \(\tfrac{1}{2} |\chi| = 3\) (Candelas+ 2018).
12/15. The tachyon